Survival via long spread in the long-range contact process

Let $q^{(t)},q^{(x)},p\in(0,1)$ be given, and let $\alpha>1$. The long-range contact process (LoRaC) is the percolation model described in the paper, with edges whose length distribution has power-law parameter $\alpha$. Survival via long spread. There exists $\alpha_c>1$ such that, for every $\alpha\in(1,\alpha_c)$, the LoRaC percolates. This conjecture asks whether survival can result from infecting sufficiently many houses rather than from infections persisting for a long time; it is presented as an open question, with a renormalisation argument suggested but not supplied.